
By theorem  REF
\(\sqrt{ \frac{Nl}{b} } \frac{U_F(b/l) -\vartheta _{N-\frac{Nl}{b}, N, T} }{\frac{b}{l}U^{\prime }_F(\frac{b}{l})} =O_p(1).\) 
Conditions of theorem REF  hold, since \(k={Nl}/{b}\sim N^{1-m}\)  and conditions of proposition REF  are assumed to hold for \(\delta =1-m\) .

Since conditions of proposition REF  hold for \(N\) , they hold for \(b\) , and lemma REF  applies. We conclude that
\(\frac{U_F(\infty )- U_F(b)}{ \vartheta _{b-q, b, T}^{(s)} - \vartheta _{b, b, T}^{(s)}} = O_p(1)\) 

Finally, by corollary 1.1.14 in [1]} under assumption REF  \({ ({b}/{l})U^{\prime }_F\left(\frac{b}{l}\right) }/{ (U_F(\infty )-U_F(b))}\rightarrow -\gamma \) . Multiplying this by \(\left(Nl/b \right)^{-1/2}\rightarrow 0\)  shows that overall the last term is \(o(1)\) .

